I finally managed to get a static musl-based toolchain! To do so, I used an alpinelinux container and compiled gcc multiple times.

I now have gcc, binutils, patch, make, m4, and so on, statically compiled.
If anyone wants to try the same, I'd suggest starting there: https://github.com/richfelker/musl-cross-make.
It will let you build a musl-based cross-compiler that's portable.

I'm now able to build packs for my distro ON my distro (currently within an LXC container). I'm facing a few errors, but they are mostly related to the tools I use, and that don't behave like GNU's. For reference, here are the "tool swap" I made to avoid GNU:

• glibc -> musl
  
• bash -> mksh
  
• coreutils -> sbase
  
• bison -> byacc
  
• flex -> re2c
  
• gawk -> bwk
  
• sed -> ssed
  

There are still some quacks, but this toolset is rather usable for now. As a matter of fact, here are the GNU/tools I'm still relying on, and the replacement I'm considering (if any):

• binutils -> elftoolchain
  
• gcc -> pcc / clang
  
• libacl
  
• libattr
  
• libmpc
  
• libncurses -> ncurses
  
• m4
  
• make
  
• patch